  Uniqueness Algorithm with Diagonal and Anti-diagonal Projections Tanuja Srivastava, Shiv Kumar Verma Department of Mathematics, Indian Institute of Technology Roorkee Roorkee, Uttarakhand-247667 tanujfma@iitr.ernet.in, shiviitr2003@gmail.com ABSTRACT The uniqueness of solution for reconstruction problem with two orthogonal projections namely in diagonal and anti-diagonal direction is discussed and an algorithm to check the uniqueness of solution is proposed. In case of unique solution this algorithm gives the solution also. Keyword: Discrete tomography; Reconstruction from projections; Binary images. Mathematics Subject Classification (2000):15A29, 15A36, 94A08 1. Introduction: In discrete tomography the reconstruction is obtained from projections in only two or four directions. These kind of problems in mathematics are referred as ill posed problems, as they do not have unique solutions [3]. Reconstructing binary matrices from given rows sums and column sums or correspondingly in discrete tomography reconstructing binary images from its horizontal (h) and vertical (v) projections, is also ill posed problem. In literature certain conditions are discussed, when these problems generate unique solutions [1,2]. To check these conditions to guarantee the unique solution is also not very easy they require algorithms. One of such algorithm in pattern recognition was given by Wang [7]. In present work, the orthogonal projections, in two directions diagonal (45 0 ) (d) and anti diagonal (135 0 ) (ad) are considered and in this case the non uniqueness or switching components [4,5] are defined and an algorithm to check presence of switching components or its complement to check the uniqueness of solution is proposed. The two directions d and ad are considered instead of hv directions, since dad projections provide better solutions than hv directions in many cases [6]. In section 2, the discrete tomography problem with dad projections is given with definition of switching component. In section 3, the proposed algorithm to check uniqueness of solution and its proof is given. The paper is concluded with validation of algorithm in some examples. International Journal of Tomography and Simulation [ISSN 2319-3336]; Year: 2013, Volume: 23, Issue Number: 2; [Formerly known as “International Journal of Tomography & Statistics” (ISSN 0972-9976; 0973-7294)]; Copyright © 2013 by CESER PUBLICATIONS www.ceser.in/ijts.html www.ceserp.com/cp-jour www.ceserpublications.com